Pith Number

pith:RTTGLDSE

pith:2026:RTTGLDSE6UPPUZXGP5ZA24MBIE

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Helmholzian Spectra of Graphs: Novel Properties

Jianfeng Wang, Lu Lu, Yi Wang, Yongtang Shi, Zoran Stani\'c

A new graph-theoretic proof confirms that the Helmholtzian matrix represents the graph Helmholtzian operator, classifying graphs with exactly two distinct eigenvalues and giving combinatorial meaning to its polynomial coefficients.

arxiv:2605.13733 v1 · 2026-05-13 · math.CO

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Record completeness

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4 Citations open

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Claims

C1strongest claim

We present a new graph-theoretic proof that the Helmholtzian matrix indeed represents the graph Helmholtzian. Our main results are as follows: (i) a classification of graphs having exactly two distinct Helmholtzian eigenvalues; (ii) the nullity of the Helmholtzian matrix; and (iii) a combinatorial interpretation of the coefficients of the Helmholtzian polynomial.

C2weakest assumption

The graph-theoretic gradient, curl, and divergence operators are defined so that their adjoints and compositions produce a well-defined Helmholtzian operator whose matrix representation is the one studied.

C3one line summary

The Helmholtzian matrix on graphs admits a classification of graphs with two eigenvalues, a formula for its nullity, and a combinatorial interpretation of its polynomial coefficients.

References

64 extracted · 64 resolved · 1 Pith anchors

[1] K. Adiprasito, J. Huh, E. Katz, Hodge Theory of Matroids, Notices Amer. Math. Soc., 64 (2017), pp. 26–30 2017

[2] K. Adiprasito, J. Huh, E. Katz, Hodge theory for combinatorial geometries, Ann. Math., 188 (2018), pp. 381–452 2018

[3] L. Bartholdi, T. Schick, N. Smale, S. Smale, Hodge theory on metric spaces, Found. Comput. Math., 12 (2012), pp. 1–48 2012

[4] Belardo, Balancedness and the least eigenvalue of Laplacian of signed graphs, Linear Algebra Appl., 446 (2014), pp 2014

[5] F. Belardo, Z. Stani´ c, T. Zaslavsky, Total graph of a signed graph, Ars Math. Contemp., 23 (2023), #P1.02 2023

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-18T02:44:16.539635Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

8ce6658e44f51efa66e67f720d71814122fce448926eba40aa2804264b646a5b

Aliases

arxiv: 2605.13733 · arxiv_version: 2605.13733v1 · doi: 10.48550/arxiv.2605.13733 · pith_short_12: RTTGLDSE6UPP · pith_short_16: RTTGLDSE6UPPUZXG · pith_short_8: RTTGLDSE

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/RTTGLDSE6UPPUZXGP5ZA24MBIE \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 8ce6658e44f51efa66e67f720d71814122fce448926eba40aa2804264b646a5b

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "e059a85da1893002ec477a7089a18840f938d413e2871a03c4fe1129856274a7",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-13T16:12:19Z",
    "title_canon_sha256": "13f9cd976c5d8964c2ccfce0a5dd75a0e83c794891a01d3214326feab0a0c6ad"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13733",
    "kind": "arxiv",
    "version": 1
  }
}